Apparatus for Demonstration of Negative Absolute Pressures and Ordinary Vapor Pressures R. P. Oordon' Columbia University, New York. NY 10027 Although 12 years have elapsed since the publication of a noteworthy article in Scientific American ( I ) on tensile strengths of liquids, students still are surprised to learn that pressures less than zero absolute are possible. Indeed, such pressures are common! Negative pressures exist in solids bearing tensile load (although pressure is not a simple number in non-isotropic systems). Also, such pressures are exerted on the sap of trees taller than 33 f t (2). The pressure of a fluid may he defined as the force exerted by the fluid, outward, upon a unit area of the wall of its container. Liquids have considerable intermolecular forces of cohesion. Thus, we see that a column of liquid, loaded as shown in Fieure 1. can he made t o exert a ~ u l l i n eforce. or hydrostatic 'tension, inward, upon its c o n t h i n g wall. ~n indicator will read a pressure (absolute, not merely . aaupe) - . less than zero. Increased loading will eventually overcome the cohesive forces, creating small cavities in the fluid. Very small cavities will collapse and disappear, but cavities greater than a critical size will crow. - . formine bubbles. hreakine the fluid column, and returning the system to positive pressure. The pressure reauired to create a snherical cavitv of radius r in a liquid whose surface free energy (surface tension) is y is given (3) by
-
-
P = 2ylr For water a t 25'C. the measured r is 72 dvneslcm (in an interface with air). If we arbitrarily take the critical v&e of r to be 3 A, we find the necessary hydrostatic pull t o be (negative) 4700 atm. If r is taken as 10 A, then P becomes (negative) 1400 atm. In addition to the arbitrary choice of r, the use of the macroscopic y is not valid for cavities of molecular dimensions. However, the results appear to he correct in .. order of magnitude. An alternate approach is to consider the "internal pressure" of the fluid. From the entropy-representation Massieu function J1, we ohtain (4) the Maxwell relation
We have devised a simple apparatus with which negative absolute pressures can he developed very readily, albeit not very reproducibly. As an additional bonus, the apparatus may he used to obtain ordinary vapor-pressure-temperature curves, extending, if desired, to over 2 atm pressure. Moreover, vapor pressures can he determined for multicomponent systems such as gasoline, whose composition would change if distillation methods were used. The basic structure shown in Figure 2 consists of a square framework of aluminum rod rigidly mounted to a central pivot, about which rotation is possible. The glass system is firmly hut resiliently attached with tape to this framework. This consists of a mercury reservoir R, with outlet tube T , a long, bent length of standard Pyrex tubing, and a cell fitted at its top with a high-quality stopcock, of the Teflon plug type, sealed with Orings. The cell jacket also has a thermometer well W and sidearms, which may he connected to a controlled-temperature circulator. The backboard is covered with millimeter graph paper marked off in centimeters. The trough at the bottom provides stability anda measure of safety in case of mercury spillage. A few holes are drilled in the backboard, A wooden peg may he placed in any one of these holes to hold the frame in various tilted positions. The mercury introduced into the reservoir should he clean and dry, and may he degassed in situ by tilting the framework while applying vacuum to T. The sample of 2-3 mL may he introduced via the stopcock's sidearm. The sample is
generally called a "thermodynamic equation of state". The infinitesimal energy U required to increase the volume V of the fluid by unit infinitesimal amount has the dimension of pressure and is called the internal pressure of the fluid. I t may he determined experimentally by measuring the change in external pressure with temperature a t constant volume. For water, the internal pressure has been found (5) to he about 16,400 atm. Thus, we should he obliged to apply a negative pressure of this magnitude in order t o overcome the internal cohesive forces of liquid water. In practice, dissolved gases, particles of dust, and irregularities of the container surface all enhance cavity nucleation and reduce the tensile stress that a liquid can bear. This is unfortunate for the design of maritime propellers, as it favors cavitation.
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F i p e 1. Conceptual apparatvs fa production of negstlve pressures
Volume 63
Number 6
June 1986
543
Figure 2. Actual apparatus, as Wcribed in this article, for prod~otimof
negative pressures.
F!gure 3. Van der Waals isotherm, extending into negative pressure reglon.
drawn in by tilting. Air bubbles are expelled by tilting. The stopcock is then closed. By raising the cell (and lowering the reservoir), the pressure on the sample fluid is decreased. When the difference in mercury levels reaches 760 mm, the sample's pressure has been reduced to zero. A further increase in the mercury level difference brings the sample into the negative (absolute) pressure ranee. ~nvariably,the first several attempts to attain negative pressures lead to frustration, as the samole column breaks kith formation of a bubble. This must be expelled via the stopcock, and another attempt made. After a few tries, sizable negative pressures can bk attained, although not nearly as large as the theoretical values, or even those which have already been obtained in practice. The largest stress recorded for water, using a centrifuge, was -227 atm at 10 'C ( I ) . In our work. usine water that had been deeassed bv hoiline but that had not &en filtered, pressures around -55 tor; were obtained at room temperature. Surprisingly, larger values could be had with ethanol: -135 torr. This value represented the limit imposed bv the dimensions of the anvaratus. Larger negativivalues should be attainable by ap&ing a vacuum to outlet T via a ballast bulb. We found. however. that the pump vihration had an adverse effect on the stabil: ity of the column of sample. It might he thought that the glass tubing could be replaced with heavy-wall flexible transparent vacuum tub in^. However, such tubing cannot be used for static vacuum, because of its permeability to air andlor the presence of volatile monomer in its wall. The existence of metastable states of negative pressure is in accordance with the van der Wads treatment of vaporliquid equilibrium. A typical isotherm is shown in Figure 3. The equilibrium transition line is d-h. Although the curve from f to g is forbidden to real systems by the condition of mechanical stabilitv . (.6). ..the metastable reeion from d to f is accessible with due care. Below e, the liquii sample sustains a tensile stress, so that its absolute pressure is negative.
where a and b are van der Wads constants. The isotherms move farther down the pressure axis as temperatures decreases. Thus, the point f,M, corresponding to the greatest negative pressure, is found by setting T = 0, at which temperature a van der Wads substance is still a liquid or a gas. This means that V~,M= b, and
"1
To locate the lowest point, f, we set the derivative equal to zero, obtaining av aV,-Bob Pf = v,3 544
Journal of Chemical Education
For water. the critical oressure is 218 atrn (n. , ., so the maximum poniible negative hydrostatic pressure is -5,900 atm. Thesame result could have been obtained from the "internal pressure" relation, because ~~
~~~~
for a van der Wads fluid at T = 0. Of course, the utility of the apparatus we have described depends on the fact that the internal cohesion of liquid metals such as mercury is far greater than that of water or organic liquids. To improve upon our results, one could degas the sample carefully, filter it to remove dust particles, pressurize the sample before reducing pressure, apply vacuum to T while isolating the apparatus from vibration, and work at lower temperatures. The aoparatus also has been used successfullv for vaoorpressurd&easurements. However, at the conc1us;on of a i m , the system should be returned to the ambient temoerature and pressure. If a bubble of air is observed, it will de necesnary to estimate its volume and correct all data ooints for its presence.
Literature cnd (1) ApfeLR. E. Sei. Anor. 1972,227(6),58. (2) Zimmuman, M. H.Sci. A m r . 1965,208(31,133. (3) Glasatone, S. "Textbook of Physical Chemiatq". 2nd 4.;Van Noatrand. Ria-, 1946: o 4%.
